# Numerical solution for Kapitza waves on a thin liquid film

## Abstract

The flow of a liquid film over an inclined plane is frequently found in nature and industry, and, under some conditions, instabilities in the free surface may appear. These instabilities are initially two-dimensional surface waves, known as Kapitza waves. Surface waves are important to many industrial applications. For example, liquid films with surface waves are employed to remove heat from solid surfaces. The initial phase of the instability is governed by the Orr–Sommerfeld equation and the appropriate boundary conditions; therefore, the fast and accurate solution of this equation is useful for industry. This paper presents a spectral method to solve the Orr–Sommerfeld equation with free surface boundary conditions. Our numerical approach is based on a Galerkin method with Chebyshev polynomials of the first kind, making it possible to express the Orr–Sommerfeld equation and their boundary conditions as a generalized eigenvalue problem. The main advantages of the present spectral method when compared to others such as, for instance, spectral collocation, are its stability and its readiness in including the boundary conditions in the discretized equations. We compare our numerical results with analytical solutions based on Perturbation Methods, which are valid only for long-wave instabilities, and show that the results agree in the region of validity of the long-wave hypothesis. Far from this region, our results are still valid. In addition, we compare our results with published experimental results and the agreement is very good. The method is stable, fast, and capable to solve initial instabilities in free surface flows.

## Keywords

Liquid film Gravity-driven flow Instability Chebyshev polynomials Galerkin method Inverse iteration method## Notes

### Acknowledgements

Bruno Chimetta is grateful to the Emerging Leaders in the Americas Program (ELAP) and to Capes for the scholarship Grants. Mohammad Hossain is grateful to Western University for providing some computational resources. Erick Franklin is grateful to FAPESP (Grant No. 2016/13474-9), to CNPq (Grant No. 400284/2016-2) and to FAEPEX/UNICAMP (Conv. 519.292) for the provided financial support.

## References

- 1.Kapitza PL (1948) Wave flow of thin layers of a viscous liquid. Part I. Free flow. Zh Eksp Teor Fiz 18(1):3Google Scholar
- 2.Kapitza PL, Kapitza SP (1949) Wave flow of thin layers of a viscous fluid. Zh Eksp Teor Fiz 19:105zbMATHGoogle Scholar
- 3.Benjamin TB (1957) Wave formation in laminar flow down an inclined plane. J Fluid Mech 2(06):554MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Yih CS (1963) Stability of liquid flow down an inclined plane. Phys Fluids 6(3):321CrossRefzbMATHGoogle Scholar
- 5.Benney D (1966) Long waves on liquid films. J Math Phys 45(2):150MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Thomas LH (1953) The stability of plane Poiseuille flow. Phys Rev 91(4):780MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Lin CC (1946) On the stability of two-dimensional parallel flows. Part III. Stability in a viscous fluid. Q Appl Math 3(4):277CrossRefzbMATHGoogle Scholar
- 8.Dolph CL, Lewis DC (1958) On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseuille flow. Q Appl Math 16:97MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Orszag SA (1971) Accurate solution of the Orr–Sommerfeld stability equation. J Fluid Mech 50(04):689CrossRefzbMATHGoogle Scholar
- 10.Floryan JM, Davis SH, Kelly RE (1987) Instabilities of a liquid film flowing down a slightly inclined plane. Phys Fluids 30(4):983CrossRefGoogle Scholar
- 11.Liu J, Paul JD, Gollub JP (1993) Measurements of the primary instabilities of film flows. J Fluid Mech 250:69CrossRefGoogle Scholar
- 12.Kalliadasis S, Demekhin E, Ruyer-Quil C, Velarde M (2003) Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J Fluid Mech 492:303MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Wierschem A, Aksel N (2003) Instability of a liquid film flowing down an inclined wavy plane. Physica D 186(3):221MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Baxter SJ, Power H, Cliffe KA, Hibberd S (2009) Three-dimensional thin film flow over and around an obstacle on an inclined plane. Phys Fluids 21(3):032102CrossRefzbMATHGoogle Scholar
- 15.Liu R, Liu Q (2009) Instabilities of a liquid film flowing down an inclined porous plane. Phys Rev E 80(3):036316CrossRefGoogle Scholar
- 16.Rohlfs W, Pischke P, Scheid B (2017) Hydrodynamic waves in films flowing under an inclined plane. Phys Rev Fluids 2(4):044003CrossRefGoogle Scholar
- 17.Drazin PG, Reid WR (2004) Hydrodynamic stability, 2nd edn. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- 18.Batchelor G (2007) An introduction to fluid dynamics. Cambridge University Press, CambridgeGoogle Scholar
- 19.Charru F (2007) Instabilités hydrodynamiques, 1st edn. EDP Sciences, Les UliszbMATHGoogle Scholar
- 20.Hossain MZ (2011) Convection due to spatially distributed heating. Ph.D. thesis, The University of Western OntarioGoogle Scholar
- 21.Trefethen LN (2000) Spectral methods in MATLAB. Siam, PhiladelphiaCrossRefzbMATHGoogle Scholar
- 22.Gaster M (1962) A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J Fluid Mech 14(2):222MathSciNetCrossRefzbMATHGoogle Scholar